The tropical Grassmannian containment conjecture for weighted trees

Let $m$ and $n$ be positive integers, let ${\mathcal{T}_{n}}$ denote the space of weighted trees on $n$ leaves, and let $\phi^{(m)}$ be the map sending a dissimilarity vector to its $m$-dissimilarity vector. Let $\mathcal{G}_{m,n}$ denote the tropical Grassmannian.

Weighted-tree tropical Grassmannian conjecture. For $m\geq 5$, one has

$$\phi^{(m)}({\mathcal{T}_{n}})\subset \mathcal{G}_{m,n}\cap \phi^{(m)}({\mathbb{R}}^{n\choose 2}).$$

The theorem proved in the paper establishes the analogous containment for $m=4$; this conjecture proposes that it remains true for all higher $m$, although the paper does not resolve the question.